# Risk bounds for reservoir computing

arXiv: https://arxiv.org/abs/1910.13886 · HTML: https://mlsanity.com/paper/1910.13886/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2019.

## Built on by (most cited first, 22 of 22)

- [Universal Approximation Under Constraints is Possible with Transformers](https://mlsanity.com/paper/2110.03303/) (2021)
- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Neural Oscillators are Universal](https://mlsanity.com/paper/2305.08753/) (2023)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)
- [A Theoretical Analysis of the Test Error of Finite-Rank Kernel Ridge Regression](https://mlsanity.com/paper/2310.00987/) (2023)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Ito Diffusions](https://mlsanity.com/paper/2105.10102/) (2021)
- [Universality and approximation bounds for echo state networks with random weights](https://mlsanity.com/paper/2206.05669/) (2022)
- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [Universal Regular Conditional Distributions](https://mlsanity.com/paper/2105.07743/) (2021)
- [State-space systems as dynamic generative models](https://mlsanity.com/paper/2404.08717/) (2024)
- [The Mori-Zwanzig formulation of deep learning](https://mlsanity.com/paper/2209.05544/) (2022)
- [Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing](https://mlsanity.com/paper/2309.04557/) (2023)
- [Towards Explainable Machine Learning: The Effectiveness of Reservoir Computing in Wireless Receive Processing](https://mlsanity.com/paper/2310.04956/) (2023)
- [Quantum Reservoir Computing and Risk Bounds](https://mlsanity.com/paper/2501.08640/) (2025)
- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [Echo State Networks as State-Space Models: A Systems Perspective](https://mlsanity.com/paper/2509.04422/) (2025)
- [Path classification by stochastic linear recurrent neural networks](https://mlsanity.com/paper/2108.03090/) (2021)
- [Towards xAI: Configuring RNN Weights using Domain Knowledge for MIMO Receive Processing](https://mlsanity.com/paper/2410.07072/) (2024)
- [QuaRK: A Quantum Reservoir Kernel for Time Series Learning](https://mlsanity.com/paper/2602.13531/) (2026)
- [Universal Approximation of Linear Time-Invariant (LTI) Systems through RNNs: Power of Randomness in Reservoir Computing](https://mlsanity.com/paper/2308.02464/) (2023)
- [Upper Generalization Bounds for Neural Oscillators](https://mlsanity.com/paper/2603.09742/) (2026)

## Abstract (arXiv)

We analyze the practices of reservoir computing in the framework of statistical learning theory. In particular, we derive finite sample upper bounds for the generalization error committed by specific families of reservoir computing systems when processing discrete-time inputs under various hypotheses on their dependence structure. Non-asymptotic bounds are explicitly written down in terms of the multivariate Rademacher complexities of the reservoir systems and the weak dependence structure of the signals that are being handled. This allows, in particular, to determine the minimal number of observations needed in order to guarantee a prescribed estimation accuracy with high probability for a given reservoir family. At the same time, the asymptotic behavior of the devised bounds guarantees the consistency of the empirical risk minimization procedure for various hypothesis classes of reservoir functionals.